Journal of Wind Engineering and Industrial Aerodynamics, 41-44 (1992) 485--496 Elsevier
485
Statistical analysis o f high return period wind speeds
b a S. Lagomarsino , G.Piccardo a and G. Solari
a Institute of Science of Constructions, University of Genoa, Via
Montallegro 1, 16145 Genoa, Italy
bDepartment of Structures, University of Calabria, 87036 Arcavacata di
Rends, Cosenza, Italy
A b s t r a c t This paper demonstrates that in the range of high return periods
different extreme wind speed distributions tend to agree or to diverge due to the parameters of the parent distribution. General criteria are there- fore established to select a priori the most reliable or prudential model. Approximate relationships are also derived, for evaluating the parameters of a given model assuming as known the parameters of another.
1. INTRODUCTION
T h e s t a t i s t i c a l a n a l y s i s o f w i n d s p e e d i n w e l l b e h a v e d c l i m a t e s i s ~radltlonally carried out in two subsequent steps. In the first step the probability distribution of the parent population is determined. In the second step the statistical analysis of extreme wind speeds is performed usually applying three different methods herein referred to as process analysis, population analysis and asymptotic analysis; this last method does not require the preliminary evaluation of the p&rent distribution.
The practical comparison of these methods reveals that, when correctly applied, they furnish almost coincident results in the range of the return periods not largely exc~edlng the number of years for which measurements are available. However, they often lead to increasing divergencies when considering return periods well above this range. This situation becomes important whenever exceptional structures are of concern. In these cases it is difficult and often impossible to establish, in general terms, which of these methods is the most reliable. In fact, from a theoretical point of view, every kind of approach has merits and defects; on the other hand, experimental judgements cannot be given, since no data base is at present
large enough to allow such evaluations. Starting from these considerations, the reciprocal limit behaviour of
extreme wind speed distributions is analyzed. In this way it is demonstrat- ed that different models tend to agree or to diverge due to the parameters of the parent distribution; general criteria are therefore established to select a priori the most reliable or prudential method. Furthermore, several approximate but conceptually consistent relationships are derived, for evaluating the parameters of a given model assuming as known the parameters of another; some formulae previously proposed in the technical
literature are in this way obtained again, other formulae are judged on the contrary to be too approximate or, in some cases, definitely unjustified.
Numeric examples are developed by means of a computer program implemented by the Authors. The data used relates to wind speeds, averaged over i0 minutes, measured by ITAV (Ispettorato delle Telecomunicazioni ed Assisten- za al Volo) in the meteorological stations of the Italian Air Force. Model parameters are evaluated, for every case, according to the most efficient estimation criterion among those discussed in Refs. [1,2].
2~ FARENT DISTRIBUTION
The use of Weibull's distribution [3] as a probabilistic model of the mean wind speed arises from pure empiricism but is by now accepted by all the sectors interested in wind phenomena [4,5,6,7]. It is defined by:
k' v k'-i v k' = -- exp[- (~7) ] fv (v) c' (~7) (v~O) (1)
v k' Fv(V ) = 1 - exp[-(~) ] (v>-O) (2)
Eqs. (1,2) coincide with the exponential distribution for k'=l, with the Raileigh's distribution for k'=2.
It must be observed that Weibull's distribution imposes conditions fv(O}= F~e(O)=O. The anemometric recordings, on the other hand, give, accordifig to
site examined, more or less long periods of wind calms. This incongru- ence can be eliminated by applying the censured technique [8] or the hybrid technique [9]. The former considers velocities v below the lower threshold v of the anemometric instrument as not reliable, modifies the values in
m
interval [0,v ) attributing an appropriately varied distribution to them; 8 e the model parameters k ,c' are regressed to modified data. The latter
accepts the instrument response substituting Eqs. (1,2} by the formulae.
k" v k"-I v k" fv(V) = (I-A) 6(v) + A c" (~) exp[-(~) ] (vi0) (3)
Fv(V) i A exp[ Z.. k" = - -(c,,) ] (v~0} (4}
in which A 18 the probability ~hat V)0, ~( ) is the Dirac's operator; parameters k",c" are estimated on the basis of the only values v)0.
It is demonstrated [2) that k"~k' and c"~c'; furthermore the smaller A, the higher k"/k' and c"/c'; naturally k'=k" and c'=c" for A=I; in this case Eqs. (3,4) coincide with Eqs. (i,2}. The application of Eqs. (1-4} to the Italian anemometric data shows that the hybrid technique is definitely better than the censored technique since it gathers, more confidently, even the values in the tails of distributions (Figs. 1}.
3. EXTREME DISTRIBUTIONS
The probability distribution of the maximum wind speed, M, over a time period T assumed as unitary, is usually calculated on the basis of three alternative methods defined in order as process analysis, population analysis and asymptotic analysis.
Process analysis treats the wind velocity according to a stochastic stationary process [10]. The average number N of the up-crossings of the threshold v in the unit time T is given by the equation [11,12]:
487
1 . E - 6 , - , 1 .E -5 > 1 .E -4
i, > 1 . E - 3 I
" - I . E - 2
_ 0.05
-J 0.1 m < m
O n-
i.i 0.5 O Z bJ C~ i,i bJ
X i,i
0 .9
~_~" I I , I , I , I I
" WEIBULL PROBABIU'W PAPER
CENSORED DISTRIBUTION A = 1., k = . 6 8 5 , c = . 7 0 6 m / s
_ - - HYBRID DISTRIBUTION A = . 329 , k = . 8 6 9 , c = 1 . 6 0 5 m / =
I I I I ' I ' 1 '
2 5 10 2 0 3 0 4 0 50
WIND SPEED v (m/s)
1 . E - 6 1 .E -5
, b > 1 . E - 4
i, > 1.E-3
I ' - - 1 . E - 2
__ 0 . 0 5
-J 0.1 m < m 0 n- Q.
L=J 0 . 5 0 Z LIJ C3 i . i LIJ C~ X bJ
0 .9
I I I I I J I l l ' .
- / PIAN ROSA'
WEIBULL P
/ / , CENSORED DISTRIBUTION + / / A = 1 . k = : .~Ts.
/ ~ / " ¢ = 6,780 rn / s "
s / / " - - - HYBRID DISTRIBUTION , s / A -- . 909 , k = 1 . 4 4 5
/ c = 7.326 m / s I I I I ' I ' I '
2 5 ~o 20 2 0 4 0 5 0
WIND SPEED v ( m / s )
Figures 1. Examples of parent probability distributions.
N(v) = J" ~, fv~(V,~,) d~ 0
(5)
in which V is the derivative process of V; fv~(v,~) is the joint density function of V and V; v and ~ are the state "v'ariables of the distribution. Eq. (5) takes a particularly simple expression assuming V and V as stati- stically independent [2,10]; in this case fv0(V,¢)= fv(V) f¢(~), f¢(~) being the density function of V. Eq. (5) for th£i reason becomes [I,2]:
~(v) = A fv(V) (6)
A = J' ~ f¢(~,) d~ 171 0
I n t r o d u c i n g t h e f u r t h e r h y p o t h e s i s t h a t v i s a s u f f i c i e n t l y h igh t h r e s h o l d , i t s u p - c r o s s i n g s can be c o n s i d e r e d as r a r e and i n d e p e n d e n t e v e n t s and treated therefore as a Polssonian process. The cumulative distribution of maximum, FM(V) 0, is in this case:
FM_A(V) = exp[-A fv(V) ] (8)
Population analysis considers data bases composed of N* values from which N samples formed by n=N*/N values corresponding to time T are extracted according to a principle equivalent to casuality. Assuming these samples as statlstically independent, the cumulative distribution of maximum, FM(V), is equal to the cumulative parent distribution, Fv(V), raised to the-h-th power. In reality, the hypothesis of statistical independence 15 wholly unacceptable. It can be removed by expressing FM(V) through formula [4,5]:
n' (9) ZM_n. (v) = [Zv(V)]
n' being the number of data effectively independent in the course of T.
488
Asymptotic analysis is the best known and most used procedure. It is demonstrated [13] that, for n' tending to infinity, the extreme distribution (9) tends, in accordance with rules associated to the tail of the parent distribution [14], to limit distributions, 8o called asymptotic. Assuming that the parent distribution i8 expressed by Weibull"s model (Eqs. 1-4) it comes into the class of the exponential tail distributions whose extreme distribution tends towards the type I asymptotic distribution:
FM_I(V ) = e x p { - e x p [ - a ( v - u ) ] } (1o)
u being the mode and 1/a the dispersion. However, since Eq. (10) i8 unlimited, it i8 not completely adequate conceptually to represent the wind speed (v~0). To thls one can add that applying the asymptotic treatment to samples composed by a finite number of independent data, errors increase on increasing the return period of the estimates concerned.
Comparing the results given by Eqs. (8,9,10) it is clear that these expressions, if correctly applied, give estimates of the maximum wind speed which are more or less coincident in the domain of the mean return period R not much greater than the time period for which records are available. On the other hand they often tend to diverge with the increase of R (Figs. 2).
60 60 ~ ' ~
PIAN ROSA'
50 50 GUMBEL PROBABILI1Y PAPER HYBRID PARENT DISTRIBUTION
,o
>
L 20
lO "i
o
I I I I I I , , , . m l ~ **1
TORINO
GUMBEL PROBABILIW PAPER CENSORED PARENT DISTRIBUTION
2 5 10 20 50 100 1000 5000 MEAN RETURN PERIOD R (yeors)
40 E
z 20
lO H./
, ASYMPTOTIC ANALYSIS (a ,~ .352, u - 28.488)
--- PROCESS ANALYSTS (X = 4200)
. . . . POPU~T=ON ANAI.YSIS ( n ' = 121s)
2 51o'2o5oloo looo sooo MEAN RETURN PERIOD R (years)
Figures 2. Examples of extreme probability distributions.
F i g s . 3 show Gumbel n o n - d i m e n s i o n a l p r o b a b i l i t y p a p e r s in which d i a g r a m s o f f u n c t i o n s v = v ( R ) , R = 1 / [ 1 - F . . ( v ) ] , x = A , n ° , I , a r e c o n s t r u c t e d based upon t h e c r i t e r ~ o n f o r m u l a t e d in ~ and t h e r e i n a p p l i e d l i m £ t e d l y t o Eqs. (9 ,10) w i th k = l , 2 ; o b s e r v e t h a t t h e e x t e n s i o n of t h e s e d i ag rams t o Eq. (8) makes them e x p l i c i t l y dependent on c; A i s assumed t o be u n i t a r y . These d i ag rams f u l l y c o n f i r m t h e d i v e r g e n t c h a r a c t e r of t h e e s t i m a t e s o f v in t h e r a n g e of t h e h igh v a l u e s o f R, e m p h a s i z i n g t h e s t r i c t d e p e n d e n c e o f t h e i r t e n d e n c i e s upon k . I t seems q u a l i t a t i v e l y c l e a r , i n p a r t i c u l a r , t h a t (10)
489
underestimates v with reference to (8,9) if k<l, while (8,9) furnish lower estimates of v compared to those given by (10) if k>l; no judgement can be expressed according to the relative tendencies of (8,9). However, it is noteworthy to make an important specification. These diagrams do not give: as seems to emerge from some of their interpretations [15], the rate of convergence, with the increase of A and n', of Eqs. (8,9} to Eq. (i0), but rather the tendency of Eqs. (8,9) to become straight lines as Eq. (10) rigorously is. In fa,ct, in Gumbel real probability papers, when increasing A and n', VA(R) and Vn, (R} respectively move in parallel to theirselves.
MEAN RETURN PERIOD R (years) 10 50 100 500 1000 I I I I J ' l
n'=1 /~ . ; - 100001 / , , , , ' ;
s > , , . .~ ; " ~-1oooo O i ~ ,. 100
/ k = 1.5 c = 5 . 0
• ASYMPTOTIC ANALYSIS
PROCESS ANALYSIS POPULATION ANALYSIS
I , ~ > 2 - 0
- 2
MEAN RETURN PERIOD R (years) lo so loo soolooo I I I I /
n ' - 1 /
V:;0060~ / / - - ~ .~ , . ' .L - . :
~ . ' ~ ~ < ~ . ,oooo
/ ~ ~X ,, 1 0 0 _
/ k = ~0
_ c = 5 , . 0
ASYMPTOTIC ANALYSIS
PROCESS ANALYSIS POPULATION ANALYSIS
Figures 3. Theoretical extreme probability distributions.
4. LIMIT TENDENCIES
A generic cumulative distribution is monotone, not decreasing, and tending to 1 with the tending to infinity of the state variable. F~ A(V), FM_n,(V), FM_I(V) thus admit the same unitary limit for v tending to
490
infinity. Transforming Eqs. (8,9,10) into equations v =v (R), R=I/[1- F M x(v) ], x=A,n',I, it is apparent that for R tending to iXfinX~ty, v tends t~-xnfinity. The first objective of the present paper is to establish whether for v tending to infinity, F M x(V) is greater, equal or less than F M (v), x,y=A,n',I: x~y; these three conditions clearly correspond in the order, for R tending to infinity, to v (R) less, equal or greater than
x • ' s v (R). The analysis is initially made by studying the recxprocal tendenc~e o~ all couplings of the three extreme distributions herein examined.
Comparing Eqs (9) and (10) one sees that F_ , (v) is less, equal or • --n
greater than Fu v(v), and therefore v_,(R} is grea~er, equal or less than vi(R ), if G* is greater, equal or less than G(v), being:
G* = exp(au) (11) An '
vk -In{l - A exp[-(c) ]}
G(v) = (12) v
A exp[- c (ac)]
It is demonstrated that G* = G(v) for any value of v and R, if and only if k=l, at=l, A-~O, exp(au)/(An')'~l. Table 1 summarizes the different situations arising for v and R tending to infinity. Figs. 4 show the diagrams corresponding to G* : G(v) in the domain of the values of most practical interest; they can be used to evaluate the values of v and R above which the limit tendencies listed in Table 1 are satisfied; it is apparent the marginal role of A especially for high values of v and R.
Comparing Eqs. (8) and (10) one sees that FM_A(V) is less, equal or greater than F M y(v), and therefore vA(R ) is grea'~er, equal or less than vI(R ) , if H* i~-~reater, equal or lesk than H(v), being
~aA (13)
k (~)k-1 e x p [ . ( ~ ) k ] H(v) = (14}
(ac) ] (ac) e x p [ - c
It is demonstrated that H* = H(v) for any value of v and R, if and only If k=l, at-l, exp(au)/(AaA)=l. Table 2 summarizes the different situations arising for v and R tending to infinity. Figs. 5 show the diagrams corresponding to H* = H(v); they can be used to evaluate the values of v and R above which the limit tendencies listed in Table 2 are satisfied.
Comparing finally Eqs. (8) and (9) one sees that F g A(V) is less, equal or greater than F ,(v), and,therefore vA(R } is gre~'er, equal or less than Vn, (R), if J~-~s greater equal or lebb than J(v), being:
A J* = ( 1 5 )
nwc
- A exp[-(v)k]} ~ln(1 %g
J(v) = (16) Ak (c)k-I exp[_(v)k]
It is demonstrated that J* = J(v) for any value of v and ~ £f and only £f k:l, A~O, A/(n'c):l. Table 3 summarizes the different ~ituationa arising for v and R tending to infinity. Fig. 6 shows the diagram~ corre~pondlng to J* = J(v); they can be used to evaluate the values of v and R above which
491
Table I , " Limit reciprocal tendencies of population and asymptotic analyses.
k<l
G ( v ) - . + =
F..,,,(v)< FM.z(v)
V,,oCR) >v= (R)
a c > I
G ( v ) - . * =
F,.,,.(v)< FM.=(v)
v.,(R) >vz (R)
e x p ( a u ) • < 1
An'
F...<v)< F..z(v)
v.,(R) >vI(R )
k = !
o c = 1
G (V) -* 1
exp(au) . " ~ ' " - - ' = 1
An'
F,.,,.(v)--F,.=(v)
vn,(R) =vz (R)
e x p ( o u ) - - - . - - - - - ~ > 1
An '
F...<v)> F..=(v)
vn.(R) <vl (R)
o ~ < 1
c (v) -. o
F..,,.(v)> F..x (v)
v..(R) < v, (R)
k > I
G (v)-,, 0
F,~<v)> F..I(v)
v..(R) <n (R)
100 ~ . ~ ~ l . ~ . ~ ~ j . ~ ~ . _
/J/ / - 20
10
. ~ , 5 .
2 J - - - - - - ' / / / / .
~ o.~ ~ ~ / .
o., t ~ . ~ . . ~ - - - o, I \\%'~X\
°"1 \ " 0,02
O O l - l ~ p ~ ~ ~ I / / 2 \ \ .5 ~0 20 50 ~00
k " - 3, / ' / ' k =, 2.~ ' k =~ ~.~,\ v/c
lOO
50
20
10
. .~ 5 c
~ 2
~ 0.5
x ~ O.2
0.1
0.05
0.02
0,01
= I I I I
1.o A = I .
k - . 5 A ~ O
k = l .
,
k - ~.s. 1 v /c
k , , . 5 k - 1,
,oo t_ .__, ~ . . ~ _ _ _ . , _ _ _ . s o ~ ~ c = 2.0
. - - . . A = 1 .
20 / - A . .~0 10
~o . , i \\\\~
°"t Ill 0,05
°°' I I I 1 0.Ol ~ ,
,, = ~ ~..~'-....~ / ,o ,o ~o lOO k = =.5:,> ' k " 2./k v / c
100
5 0
20
10
.'L-" 5 r-
~ 2
o 0,5 o . x ~ 0,2
0,1
0,05
0.02
0.01
k - .5, k ,, 1. /
k - 2 . 5 j ~
k = , 3 .
1 2 5 10 20
v / c
oc = 5.0 A = I
A ~ 0
k = 1.5
50 100
Figures 4. Parametric diagrams of G* = G(v).
100
50
Table 2. Limit reciprocal tendencies of process and asymptotic analyses.
k < 1
H (v) .+ +==
F=.x(v)< F,.i(v)
v x (R) > vl (R)
a c > 1
H (v) -+ +==
F..x(v)< F..,(v)
v>.CR)>vl (R)
exp(ou) ~ < 1
),oA
F..x(v)< %.,(v)
v~,(R)>vÁ (R)
k = 1
O C : I
H ( v ) = 1
exp(au)
),aA
F,.x(v)=Fl,.i(v)
vx(R)--vl (R)
"V" v,R
expCou) ~ > 1
F=.x(,)> F=.i(v)
v~(R)<vl (R)
o c < I
H (v) + O
F=.x(v)> %.,(v)
v~ (R) <v! (R)
k > l
H (v) '+ 0
Fu.x(v)> Fu.,(v)
vx(R)<vz(R)
~oo ~ , ~ , ~ , ~ , ~ , ~ ~ _ ]j/' / --.-- 10
k =, .5 = - K ~ / / / ~ -
o: t ~ - - j - y - - t - - -
o:, t //\\%\)~X~',~< -~-~- - o.o,1 -
0,02
o,o, + . _ ~ 1 i l l ~ \5 ~o =o 5o ~oo
k iB , ~ ~ / " ._,,; vto
~oo
:~ / / / / / o c = 1.0
,o , - + d / / / " i Y ~ 2 ,.~ 1 / / k == 1.
~ 0.5
~, o.= ~\%\\\\ \ \
0,05
0,02
0,01 . / / i I 5 to 20 so Ioo
k - , ~ _ v / c
k l , 5 k - 1,
, , \ , / , , ,
= 2,0 20 10
r< v 2
492
0,5
I, 0,1
0,05
0,02 0,01
~ ~ ~ ' i ' ' k = 10 20
==
!
50 100
k - ,S, k - 1, I
100
= 1 2O
o 0,5 1
~ 0,2 kin2. ~ I t[ 0,1 k = 2,5...."
0,05 k = 3,
0,02
o,of 1 2 5 10 20
vlc
= =
ac = 5.0 -
k - 1,~
50 100
Figures 5. Parametric diagrams of H ~ ffi H(v) .
493
Table 3. Limit reciprocal tendencies of process and population analyses.
k < 1
,J (v) ~ , + -
F..,(v)> F~,,.(v)
v,(R)<v..(R)
< 1 ftt'¢
v..,(v)> v~.(v)
v,(e)<v,,.(e)
k=l
a (v) -.,,1
'X' X - - " 1 > 1 N'C N'C ~,,,
r,.x(v)=F,.,.(v) Fu.x0,)< F~,(v) v x (R ) - vn . (R ) v x (R) > v.,(R)
k > 1 , .
J (v)-.O
F.~(v)<V....(v) ,,,(R)>v,,.(R)
100
50
20
10
5
25 , ~ 0,5
0.2
O.1 -, 1,5
O,05
0.02
o.oi
1 2 5 lo 20 50 100
v/c
Figure 6. Parametric diagrams of J* = J(v).
the limit tendencies listed An Table 3 are satisfied; At is apparent, as in Figs. 4, the marginal role of A especially for high values of v and R.
Joining together the above flared reaults, it ie apparent the existence of three dlstinat behaviours, firat of all associated with the value of kz Ca} if k=l, thenz
(l) at=l, exp(au)=kn' imply FM.n.(V)=FM.z(v), vn,(R)=vi(R), for large veR;
(2) at=l, exp(au)=Aak imply FM_~(v)=FM_T(V), vA(R)=vI(R), for any v,R; (3) A=n'c implies F, ~(v)=F, ~,'(~}, ~(R)=Vn0(~}, for large v,R; in all other case~'~e l~m~ conditions listed in Tables 1,2,3 come into force, these having, however, a very limited importance, k=l being a bifurcation point;
(b) if k>l, this being the case of major interest, then, for large v,R:
FM.I(V } < FM_A(v } < FM.n,(V) (17a)
v 0(R) < vA(R ) < vI(R) (17b)
for any value of all parameters. In this situation Eq. (10) gives, An relation to the Eqs. (8,9) of higher level, more prudential estimates of v the more k>l; Eq. (8) is more prudential than Eq. (9);
494
(c) if k<l, then, for large v,R:
FM_n,(V) < FM_A(V ) < FM_I(V) (18a)
v I(R) < VA(R) < Vn,(R} (18b)
for any value of all parameters. In this situation Eq. (10) gives, in relation to the Eqs. (8,9) of higher level, less conservative estimates of v the more k<l; Eq. (9) is more prudential than Eq. (8).
These considerations clearly explain Figs. 2. Remembering that the above limit tendencies are practically invariant
with respect to A, it is clear that the use of the cenqured model (1,2} in- stead of the hybrid model (3,4) has no influence on the limit behaviour of the extreme distributions. What takes on instead an essential role is that the use of Eqs. (1,2) gives rise to k values, k', less than or equal to the k values, k", given by Eqs. (3,4) . From this it follows that extreme esti- mates by Eqs. (3,4), apart from being decidedly better, are also lower than estimates based on Eqs. (1,2), independently of using Eq. (8) or (9).
It is relevant to observe that k<l does not represent a case of pure theoretical interest. In Italy, for instance, the 15% of data collected by meteorological stations exhibits k<l if fitted by the hybrid technique; this percentage reaches the 35% when using the censured model.
5. APPROXIMATE RELATIONSHIPS
Having established the criteria to select a priori the most reliable or prudential extreme distribution, the opportunity becomes obvious of instituting simple but effective approximate relationships which permit the determination of the parameters of a given model on the basis of the parameters of another model. The assigning parameters A and n' (of complex evaluation [1,2]) is clearly important in terms of parameters A,k,c,u,a (easier to be estlmated)I n' expresses the number of the independent repetitions of the mean wind speed during T and thus plays a central role in the sector of reliability analyses; the knowledge of A, instead, makes it possible to extend the process analysis from the study of the wind to t h e s t u d y of i t s e f f e c t s , accord ing t o g l o b a l r i s k e v a l u a t i o n s [16] . Also t he c a l c u l a t i o n of A from n ' , and v i c e v e r s a , can be r e l e v a n t (10) .
The problem i s fo rmu la t ed here comparing coup les of d i f f e r e n t models . The relationships linking the parameters of one to the parameters of another are based on three hypotheses: (a) the extreme distributions are more or less overlaid in the domain of the experimental data; (b) they coincide exactly for v=u; (c) u is much greater than c.
Comparing Eqs. (9) and (i0) it follows that:
1. I n # i (19)
I - Fv(U ) A e x p [ - ( u ) k]
This same comparison is formulated in [4] with the intention of evaluating u,a and therefore ~=ua in terms of k,c,n' being k=l. Generalizing these expressions to the case A~I [17], one obtains:
u = c [In(n'A)] I/k (20)
k 1-1/k a = - [ l n ( n ' A ) ] ( 2 1 ) c
495
= ua : k ln(n'A) (22)
Eq. (20) corresponds to imposition F.. , (u):FM_ (u); Eq. (21) requests the M-n -I
additional condition f_ , (u)=f. _(u), fM(v)-])eing the density function of M-- M - Z
maximum M. In the light o~ this concept, The inversion of Eq. (22) [5,17]:
1 n" = (23) ua
A e x p ( - - ~ )
i s thoroughly unjustified since it presupposes a criterion definitely up- rooted from c o n d i t i o n s F . _ . ( u } : F M ~ ( : ) e f M _ n . ( U ) : f M _ i ( u ) ; these equa t i ons are in fact clearly confl~c~ing, only exception o= case k=l, ac=l, when applied both for determining the single parameter n'.
Comparing Eqs. (8) and (10) it follows that:
1 1 A - - ( 2 4 )
fvlu) A k c k-i u k c ( ) e x p [ - ( c ) ]
Finally, comparing Eqs. (8) and (9) it follows that:
A i - Fv(U ) c (c)l-k
n- ~ = f v ( U ) = ~. (25)
Eq. (25) can be r e w r i t t e n i n t e r m s o f p a r a m e t e r s A , k , c by s u b s t i t u t i n g Eqs . ( 2 0 , 2 1 ) i n t o Eq. ( 2 5 ) . One h a s :
A c l / k - 1 n' k [In(n'A)] (26)
It is relevant to notice that applylng Eqs. (25,26), if k<l, then F M n,(V}< F_ _(v) and v ,(R)>v_(R) for any v>u; analogously, if k>l, then F~-_(v)<
M-A n A > M-A F_ ,(v) and v_(R)>v ,(R) for any v u. It is finally interesting to o~serve t~a~ Eqs . ( 25 ,~6 ) , t~oroughly deny the following formula derived from (10 ] :
A _ _ . S ( 2 7 ) n' k
However, it coincides with Eq. (25) when k-l, thle being the case in which all approximate assumptions made in (10] become rigorously exact.
Table 4 summarizes mean values, standard deviations, maxima and minima of percent errors associated with formulae in thls paragraph when applied to the 40 Italian meteorological stations studied in [18]; values between parentheses correspond to the same data base, having excluded the stations
T a b l e 4. E r r o r s a s s o c i a t e d t o a p p r o x i m a t e r e l a t i o n s h i p s .
1.2 Z 16.5 • 20.5 Z 351.4 X 7.2 • 16.1 Z 15.0 x 50.9 x (1.5 x) (11.5 x) (17.7 x) (45.8 x) (5.8 x) (15,7 x) (15.4 s) (51.9 x)
5.1 S 40.5 S 65.3 S 1925.7 S 10.6 • 37.1 ~li 56.3 S 176.0 S (5.1 s) (7.4 s) (55.3 s) (69.5 x) (10.8 s) (57.1 s) (56.5 s) (176.0 s)
i
-26.0 X -2.4 X -37.7 ~ -28.0 X -04.6 X -22.0 X -19.8 Z -19.4 X -45.4 Z (-26.0 Xl (-2.4 X) (-37.7 ~) (-8.7 Z) (-94.6 Z)I(-14.4 X) (-19.8 X) (-19.4 ~i) (-45.4 X)
496
of Cozzo Spadaro, Sanremo and Venice. It is apparent the possibility of di- viding the above relationships into three groups with different properties: (a) Eqs. (19), (20}, (24), (25), (26) give excellent approximations being only based on condition F. (u)=F. (u}, x,y=A,n',I, x#y; (b) Eqs. (21), (22) lead ~oXrough~-~ estimates involving the additional
condition f (u)=f. (u), x,y=A,n',I, x#y; -x i~y Eq. 23 are thoroughly misleading; observe in (c) Eq. (27) an~ espec ( )
particular that Eq. (23) leads in some cases (the three stations quoted above) to exceptionally high errors.
These results are definitely independent of the values assumed by k.
6 . CONCLUSIONS AND P R O S P E C T S
This paper has framed the limit tendencies of the best known models for representing extreme wind velocities. In particular it has demonstrated that these tendencies strictly depend on the k parameter of the parent distribution. If k>l the asymptotic analysis furnishes results on the safe side, while the process analysis is, with respect to the population analy- sis, more prudential. On the other hand, if k<l the asymptotic analysis becomes unconservative, while the population analysis is, at the same time, the most prudential and reliable one. A general frame has also been given of the most suitable approximate relationships aimed at estimating the pa- rameters of a given model assuming as known the parameters of another. In light of the results obtained, the opportunity clearly emerges of going deeper into this subject by further considering the directional and season- al effects as well as analyzing the role of statistical uncertainties.
,'7. REFERENCES
9 10 11 12 13 14 15 16
17 18
S. Lagomareino, G. Piccardo and G. Solar£, Proc . , ! Convegno Nazionale d i Ingegner ia de l Vento, F lorence, I t a l y (1990) 209. S. Lagomarsino, G. Piccardo and G. S o l a r i , i n p r e p a r a t i o n . W. Weibu11, J. Appl . Mech., 18 (1951) 293. A.G. D a v e n p o r t , P r o c . , I n t . Con f . on Wind E f f e c t s on B u i l d i n g s and S t r u c t u r e s , Ottawa, Canada, 1 (1968), 19. N.J. Cook, J. Wind Engng. Ind. Aerod. , 9 (1982) 295. J .P. Hanneseey, J. Appl . Meteor . , 16 (1977) 119. C.G. J e s t u s , W.R. H a r g r a v e s , A. Mikha£1 and D. G r a b e r , J . A p p l . Meteor . , 17 (1978) 350. K. Conradsen, L.B. N ie lsen and L.P. Prahm, J. C1£m. Appl . Meteor . , 23 (1984) 1173. E.S. Takle and J.M. Brown, J. Appl . Meteor . , 17 (1978) 556. L. Gomes and B.J. r i c k e t y , J. Ind . Aerod. , 2 (1977) 21. S.O. Rice, B e l l . Syst . Techn. J . , 23 (1944) 282. S.O. Rice, B e l l . Syst . Techn. J . , 24 (1945) 46. R.A. F isher and L.H.C. T i p p e t t , Proc. Camb. P h i l . Soc., 24 (1928) 180. E.J. Gumbel, Statistics of extremes, Columb£a Univ., New York (1958). R.V. Milford, J. Wind Engng. Ind. Aerod., 25 (1987) 163. A.G. Davenport, Proc., 2nd Int. Conf. on Structural Safety and Reliability, Munich, Germany (1977) 511. R.I. Harris, J. Wlnd Engng. Ind. Aerod., 9 (1982) 343. G. Ballio, S. Lagomarsino, G. Piccardo and G. Solari, Costruzioni Metalliche (1991) 147, 209.
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